Squares, Right Triangles, and Areas To learn how areas of figures can be found by decomposing the figures into triangles and rectangles or by removing triangles from larger rectangles To lay the groundwork for understanding the Pythagorean Theorem by learning that if the area of a square is s, then its side length is .

Find the area of each shape on the grid at right. The rectangle has an area of 3 square units. The area of the triangle is half the area of the rectangle, so it has an area of 1.5 square units.

You can often draw or visualize a rectangle or square related to an area to help you find the area. Find the area of square ABCD.

What’s My Area? Copy these shapes onto graph paper. Work with a partner to find the area of each figure.

If you know the side length, s, of a square, then the area of the square is s2. Likewise, if you know that the area of a square is s2, then the side length is , or s. So the square labeled d in Step 1, which has an area of 2, has a side length of units.

What are the area and side length of the square labeled e? What are the area and side length of each of these squares (a and b)?

Shown at the right are the smallest and largest squares with grid points for vertices that can be drawn on a 5- by-5 grid. Draw at least five other different-size squares on a 5-by-5 grid. They may be tilted, but they must be square, and their vertices must be on the grid. Find the area and side length of each square.

Example C Draw a line segment that is exactly units long. A square with an area of 10 square units has a side length of 10 units. Ten is not a perfect square, so you will have to draw this square tilted. Start with the next largest perfect square—that is, 16 square units (4-by-4)—and subtract the areas of the four triangles to get 10. Here are two ways to draw a square tilted in a 4-by-4 square. Only the square on the left has an area of 10 square units.